Download Identifying Integers and their Opposites

LESSON
1.1
?
Identifying Integers
and Their Opposites
Number and
operations—6.2.B
Identify a number, its
opposite, and its absolute
value.
ESSENTIAL QUESTION
How do you identify an integer and its opposite?
EXPLORE ACTIVITY 1
6.2.B
Positive and Negative Numbers
Positive numbers are numbers greater than 0. Positive
numbers can be written with or without a plus sign;
for example, 3 is the same as +3. Negative numbers
are numbers less than 0. Negative numbers must always
be written with a negative sign.
-5 -4 -3 -2 -1
The number 0 is
neither positive
nor negative.
0 1 2 3 4 5
Negative integers
Positive integers
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The elevation of a location describes its height above or below sea level,
which has elevation 0. Elevations below sea level are represented by negative
numbers, and elevations above sea level are represented by positive numbers.
A The table shows the elevations of several locations in a state park.
Graph the locations on the number line according to their elevations.
Location
Elevation (ft)
Little
Butte
A
Cradle
Creek
B
Dinosaur
Valley
C
Mesa
Ridge
D
Juniper
Trail
E
5
-5
-9
8
-3
-1 -09 -8 -7 -6 -5 -4 -3 -2 -1
0 1 2 3 4 5 6 7 8 9 1 0
B What point on the number line represents sea level?
C Which location is closest to sea level? How do you know?
D Which two locations are the same distance from sea level? Are these
locations above or below sea level?
E Which location has the least elevation? How do you know?
Lesson 1.1
7
EXPLORE ACTIVITY 1 (cont’d)
Reflect
1.
Analyze Relationships Morning Glory Stream is 7 feet below sea
level. What number represents the elevation of Morning Glory Stream?
2.
Multiple Representations Explain how to graph the elevation
of Morning Glory Stream on a number line.
EXPLORE ACTIVITY 2
6.2.B
Opposites
Two numbers are opposites if, on a number line, they are
the same distance from 0 but on different sides of 0. For -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
example, 5 and -5 are opposites. 0 is its own opposite.
Remember, the set
of whole numbers is
Integers are the set of all whole numbers and their opposites.
0, 1, 2, 3, 4, 5, 6, ...
On graph paper, use a ruler or straightedge to draw a number line. Label
the number line with each integer from -10 to 10. Fold your number
line in half so that the crease goes through 0. Numbers that line up after
folding the number line are opposites.
A Use your number line to find the opposites of 7, -6, 1, and 9.
C What is the opposite of the opposite of 3?
Reflect
8
3.
Justify Reasoning Explain how your number line shows that 8 and -8
are opposites.
4.
Multiple Representations Explain how to use your number line to find
the opposite of the opposite of -6.
Unit 1
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B How does your number line show that 0 is its own opposite?
Integers and Opposites
on a Number Line
Positive and negative numbers can be used to represent real-world quantities.
For example, 3 can represent a temperature that is 3 °F above 0. -3 can
represent a temperature that is 3 °F below 0. Both 3 and -3 are 3 units from 0.
EXAMPL 1
EXAMPLE
6.2.B
Sandy kept track of the weekly low temperature in her town for several
weeks. The table shows the low temperature in °F for each week.
Week
Temperature (°F)
Week 1
Week 2
Week 3
Week 4
-1
3
-4
2
A Graph the temperature from Week 3 and its opposite on a
number line. What do the numbers represent?
STEP 1
STEP 2
1
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STEP 1
Graph the value from Week 1 on the number line.
The value from Week 1 is -1.
STEP 2
Graph the opposite of -1.
The opposite of -1 is 1.
STEP 3
Graph the opposite of 1.
The opposite of 1 is -1.
-6 -5 -4 -3 -2 -1
Use this space to
take notes as you
5
Graph the opposite of -4.
Graph a point 4 units above 0.
B The value for Week 5 is the opposite of the opposite of the
value from Week 1. What was the low temperature in Week 5?
My Notes
6
4
-4 represents a temperature that is 4 °F below 0
and 4 represents a temperature that is 4 °F above 0.
my.hrw.com
listen in class.
Graph the value from Week 3 on the number line.
The value from Week 3 is -4.
Graph a point 4 units below 0.
The opposite of -4 is 4.
Math On the Spot
3
2
0
-1
-2
-3
-4
-5
-6
0 1 2 3 4 5 6
The opposite of the opposite of -1 is -1.
The low temperature in Week 5 was -1 °F.
Reflect
5.
Analyze Relationships Explain how you can find the opposite of the
opposite of any number without using a number line.
Lesson 1.1
9
YOUR TURN
Graph the opposite of the number shown on each number line.
Personal
Math Trainer
6.
Online Assessment
and Intervention
my.hrw.com
7.
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
0 1 2 3 4 5 6 7 8 9 10
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
0 1 2 3 4 5 6 7 8 9 10
Write the opposite of each number.
Math Talk
8.
10
9.
-5
10.
0
Mathematical Processes
11.
Explain how you could
use a number line to find
the opposite of 8.
What is the opposite of the opposite of 6?
Guided Practice
1. Graph and label the following points on the number line.
(Explore Activity 1)
a. -2
b. 9
c. -8
-10-9 -8 -7 -6 -5 -4 -3 -2 -1
d. -9
e. 5
f.
8
0 1 2 3 4 5 6 7 8 9 10
2.
3.
4.
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
0 1 2 3 4 5 6 7 8 9 10
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
0 1 2 3 4 5 6 7 8 9 10
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
0 1 2 3 4 5 6 7 8 9 10
Write the opposite of each number. (Explore Activity 2 and Example 1)
5. 4
6. -11
8. -3
9. 0
?
?
ESSENTIAL QUESTION CHECK-IN
11. Given an integer, how do you find its opposite?
10
Unit 1
7. 3
10. 22
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Graph the opposite of the number shown on each number line.
(Explore Activity 2 and Example 1)
Name
Class
Date
1.1 Independent Practice
Personal
Math Trainer
6.2.B
my.hrw.com
Online
Assessment and
Intervention
12. Chemistry Atoms normally have an electric charge of 0. Certain
conditions, such as static, can cause atoms to have a positive or a
negative charge. Atoms with a positive or negative charge are called ions.
Ion
Charge
A
B
C
D
E
-3
+1
-2
+3
-1
a. Which ions have a negative charge?
b. Which ions have charges that are opposites?
c. Which ion’s charge is not the opposite of another ion’s charge?
Name the integer that meets the given description.
13. the opposite of -17
14. 4 units left of 0
15. the opposite of the opposite of 2
16. 15 units right of 0
17. 12 units right of 0
18. the opposite of -19
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19. Analyze Relationships Several wrestlers are trying to lose weight for a
competition. Their change in weight since last week is shown in the chart.
Wrestler
Tino
Victor
Ramsey
Baxter
Luis
Weight Change
(in pounds)
-2
6
2
5
-5
a. Did Victor lose or gain weight since last week?
b. Which wrestler’s weight change is the opposite of Ramsey’s?
c. Which wrestlers have lost weight since last week?
d. Frankie’s weight change since last week was the opposite of Victor’s.
What was Frankie’s weight change?
e. Frankie’s goal last week was to gain weight. Did he meet his goal? Explain.
Lesson 1.1
11
Find the distance between the given number and its opposite
on a number line.
20. 6
21. -2
22. 0
23. -7
24. What If? Three contestants are competing on a trivia game show.
The table shows their scores before the final question.
a. How many points must Shawna earn for her score to be the opposite
of Timothy’s score before the final question?
b. Which person’s score is closest to 0?
Contestant
Score Before
Final Question
Timothy
-25
Shawna
18
Kaylynn
-14
c. Who do you think is winning the game before the final question?
Explain.
FOCUS ON HIGHER ORDER THINKING
Work Area
25. Communicate Mathematical Ideas Which number is farther from 0
on a number line: -9 or 6? Explain your reasoning.
27. Critique Reasoning Roberto says that the opposite of a certain integer
is -5. Cindy concludes that the opposite of an integer is always negative.
Explain Cindy’s error.
28. Multiple Representations Explain how to use a number line to find the
opposites of the integers 3 units away from -7.
12
Unit 1
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26. Analyze Relationships A number is k units to the left of 0 on the
number line. Describe the location of its opposite.